Simplify the expression: ${({x^3})^4}$

Hint:We have to simply solve the index (power) which is placed over the algebraic element so after simplifying the index (which is raised to power 3 in this case) we multiply it with the index placed over the bracket of the given expression using the property of indices.

Complete solution step by step:
Firstly, we write the given expression
${({x^3})^4}$

We can see that the algebraic expression has an element inside the bracket i.e. $x$ raised to its power three. So we have to use the indices to simplify the solution.

Now to solve this expression we use the index properties i.e.
${({p^q})^r} = {({p^r})^q} = {p^{q \times r}}$

So we can see that when two powers are raised to a number and separated by a bracket then after simplification the two powers are multiplied with each other.

Using this property and taking first and third part of it, we solve our expression
${({x^3})^4} = {x^{3\; \times \;4}} = {x^{12}}$

So this is the simplified form of the given algebraic expression.

Additional information: We can check this expression by putting a random value of $x$ and then applying the index formula to the expression:

Take $x = 2$

${({2^3})^4} = {(2 \times 2 \times 2)^4} = {(8)^4} = 64 \times 64 = 4096$

Now we use the simplified value of the expression and put the same value of $x$ so we have

${(2)^{12}} = (2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times
2 \times 2) = 4096$

As we can see we have got the same result by putting a random value in the expression.

Note: In layman terms, index is the ‘n-th’ power raised to a number and after simplification it is multiplied n-times by itself. After solving, the index (power) of a number will be the raised value of the number.